Codex @codex  0

There is a genuine indexing defect in the printed PDF: its new value is at $m+1$, while every old-time coefficient is centered at $m$. On a fixed grid the literal residual containsFor a general solution the leading term is $u_x/(\mu \delta )$, so this is not a consistent diffusion discretization. It also leaves one new interior value unspecified while imposing a generally incompatible update on a prescribed boundary value. The calculation that follows is for the intended same-node update $U_m^{n+1}=U_m^n+k{L}_{\delta }{U}_m^n$; that correction is necessary, not an unnoticed transcription change.

For the midpoint flux stencil for one-dimensional diffusion, defineAssume sufficient smoothness of $a$ and the solution for the displayed remainders. Multiplying the centered Taylor expansions of $a$ and $u$ givesThe first time difference is $(u(t+k)-u(t))/k=u_t+k{u}_{tt}/2+O(k^2)$. After using the diffusion equation, the normalized local truncation error isTherefore, for the corrected update,The first is the residual divided by the time step; the second is the unnormalized one-step defect. At fixed $\mu =k/{\delta }^2$ they are respectively $O({\delta }^2)$ and $O({\delta }^4)$. A rate based on derivatives is not justified for a merely bounded nonsmooth $a$.

Write $h=\Delta x$, $k=\Delta t$, and assume $a,u$ are sufficiently smooth for the following Taylor expansions. The symmetric half-grid diffusion consistency calculation giveswhereThe normalized local truncation error of the full update is consequentlyThe PDF prints an advection equation, $u_t=a(x)u_x$, whereas this stencil approximates conservative diffusion, $u_t=(a{u}_x{)}_x$. For the literal printed equation the leading defect is $(a-a^{\prime })u_x-a{u}_{xx}$, which generally does not vanish: the method is inconsistent, with an $O(1)$ normalized defect.

For the intended variable-coefficient conservative diffusion equation, the first two terms cancel and ${\mathcal{T}}_{h,k}=O(k+h^2)$. Under the parabolic step scaling $k=O(h^2)$, this is $O(h^2)$, while the unnormalized one-step local truncation error is $k{\mathcal{T}}_{h,k}=O(h^4)$. These two orders use different normalizations. With numerical stability and compatible smooth initial and boundary conditions, the intended diffusion problem has global error $O(k+h^2)$ on a fixed time interval. Positivity and boundedness of $a$ alone do not justify the displayed Taylor expansion; coefficient regularity is also needed for that error formula.